76,739 research outputs found

    On the minor-minimal 2-connected graphs having a fixed minor

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    Let H be a graph with κ1 components and κ 2 blocks, and let G be a minor-minimal 2-connected graph having H as a minor. This paper proves that |E(G)|-|E(H)|≤α(κ 1-1)+β(κ2-1) for all (α,β) such that α+β≥5,2α+5β≥;20, and β≥3. Moreover, if one of the last three inequalities fails, then there are graphs G and H for which the first inequality fails. © 2003 Elsevier B.V. All rights reserved

    Large induced subgraphs via triangulations and CMSO

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    We obtain an algorithmic meta-theorem for the following optimization problem. Let \phi\ be a Counting Monadic Second Order Logic (CMSO) formula and t be an integer. For a given graph G, the task is to maximize |X| subject to the following: there is a set of vertices F of G, containing X, such that the subgraph G[F] induced by F is of treewidth at most t, and structure (G[F],X) models \phi. Some special cases of this optimization problem are the following generic examples. Each of these cases contains various problems as a special subcase: 1) "Maximum induced subgraph with at most l copies of cycles of length 0 modulo m", where for fixed nonnegative integers m and l, the task is to find a maximum induced subgraph of a given graph with at most l vertex-disjoint cycles of length 0 modulo m. 2) "Minimum \Gamma-deletion", where for a fixed finite set of graphs \Gamma\ containing a planar graph, the task is to find a maximum induced subgraph of a given graph containing no graph from \Gamma\ as a minor. 3) "Independent \Pi-packing", where for a fixed finite set of connected graphs \Pi, the task is to find an induced subgraph G[F] of a given graph G with the maximum number of connected components, such that each connected component of G[F] is isomorphic to some graph from \Pi. We give an algorithm solving the optimization problem on an n-vertex graph G in time O(#pmc n^{t+4} f(t,\phi)), where #pmc is the number of all potential maximal cliques in G and f is a function depending of t and \phi\ only. We also show how a similar running time can be obtained for the weighted version of the problem. Pipelined with known bounds on the number of potential maximal cliques, we deduce that our optimization problem can be solved in time O(1.7347^n) for arbitrary graphs, and in polynomial time for graph classes with polynomial number of minimal separators

    Complexity of Grundy coloring and its variants

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    The Grundy number of a graph is the maximum number of colors used by the greedy coloring algorithm over all vertex orderings. In this paper, we study the computational complexity of GRUNDY COLORING, the problem of determining whether a given graph has Grundy number at least kk. We also study the variants WEAK GRUNDY COLORING (where the coloring is not necessarily proper) and CONNECTED GRUNDY COLORING (where at each step of the greedy coloring algorithm, the subgraph induced by the colored vertices must be connected). We show that GRUNDY COLORING can be solved in time O(2.443n)O^*(2.443^n) and WEAK GRUNDY COLORING in time O(2.716n)O^*(2.716^n) on graphs of order nn. While GRUNDY COLORING and WEAK GRUNDY COLORING are known to be solvable in time O(2O(wk))O^*(2^{O(wk)}) for graphs of treewidth ww (where kk is the number of colors), we prove that under the Exponential Time Hypothesis (ETH), they cannot be solved in time O(2o(wlogw))O^*(2^{o(w\log w)}). We also describe an O(22O(k))O^*(2^{2^{O(k)}}) algorithm for WEAK GRUNDY COLORING, which is therefore \fpt for the parameter kk. Moreover, under the ETH, we prove that such a running time is essentially optimal (this lower bound also holds for GRUNDY COLORING). Although we do not know whether GRUNDY COLORING is in \fpt, we show that this is the case for graphs belonging to a number of standard graph classes including chordal graphs, claw-free graphs, and graphs excluding a fixed minor. We also describe a quasi-polynomial time algorithm for GRUNDY COLORING and WEAK GRUNDY COLORING on apex-minor graphs. In stark contrast with the two other problems, we show that CONNECTED GRUNDY COLORING is \np-complete already for k=7k=7 colors.Comment: 24 pages, 7 figures. This version contains some new results and improvements. A short paper based on version v2 appeared in COCOON'1

    Densities of Minor-Closed Graph Families

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    We define the limiting density of a minor-closed family of simple graphs F to be the smallest number k such that every n-vertex graph in F has at most kn(1+o(1)) edges, and we investigate the set of numbers that can be limiting densities. This set of numbers is countable, well-ordered, and closed; its order type is at least {\omega}^{\omega}. It is the closure of the set of densities of density-minimal graphs, graphs for which no minor has a greater ratio of edges to vertices. By analyzing density-minimal graphs of low densities, we find all limiting densities up to the first two cluster points of the set of limiting densities, 1 and 3/2. For multigraphs, the only possible limiting densities are the integers and the superparticular ratios i/(i+1).Comment: 19 pages, 4 figure

    Beyond Bidimensionality: Parameterized Subexponential Algorithms on Directed Graphs

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    We develop two different methods to achieve subexponential time parameterized algorithms for problems on sparse directed graphs. We exemplify our approaches with two well studied problems. For the first problem, {\sc kk-Leaf Out-Branching}, which is to find an oriented spanning tree with at least kk leaves, we obtain an algorithm solving the problem in time 2O(klogk)n+nO(1)2^{O(\sqrt{k} \log k)} n+ n^{O(1)} on directed graphs whose underlying undirected graph excludes some fixed graph HH as a minor. For the special case when the input directed graph is planar, the running time can be improved to 2O(k)n+nO(1)2^{O(\sqrt{k})}n + n^{O(1)}. The second example is a generalization of the {\sc Directed Hamiltonian Path} problem, namely {\sc kk-Internal Out-Branching}, which is to find an oriented spanning tree with at least kk internal vertices. We obtain an algorithm solving the problem in time 2O(klogk)+nO(1)2^{O(\sqrt{k} \log k)} + n^{O(1)} on directed graphs whose underlying undirected graph excludes some fixed apex graph HH as a minor. Finally, we observe that for any ϵ>0\epsilon>0, the {\sc kk-Directed Path} problem is solvable in time O((1+ϵ)knf(ϵ))O((1+\epsilon)^k n^{f(\epsilon)}), where ff is some function of \ve. Our methods are based on non-trivial combinations of obstruction theorems for undirected graphs, kernelization, problem specific combinatorial structures and a layering technique similar to the one employed by Baker to obtain PTAS for planar graphs

    Complexity of the Steiner Network Problem with Respect to the Number of Terminals

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    In the Directed Steiner Network problem we are given an arc-weighted digraph GG, a set of terminals TV(G)T \subseteq V(G), and an (unweighted) directed request graph RR with V(R)=TV(R)=T. Our task is to output a subgraph GGG' \subseteq G of the minimum cost such that there is a directed path from ss to tt in GG' for all stA(R)st \in A(R). It is known that the problem can be solved in time V(G)O(A(R))|V(G)|^{O(|A(R)|)} [Feldman&Ruhl, SIAM J. Comput. 2006] and cannot be solved in time V(G)o(A(R))|V(G)|^{o(|A(R)|)} even if GG is planar, unless Exponential-Time Hypothesis (ETH) fails [Chitnis et al., SODA 2014]. However, as this reduction (and other reductions showing hardness of the problem) only shows that the problem cannot be solved in time V(G)o(T)|V(G)|^{o(|T|)} unless ETH fails, there is a significant gap in the complexity with respect to T|T| in the exponent. We show that Directed Steiner Network is solvable in time f(R)V(G)O(cgT)f(R)\cdot |V(G)|^{O(c_g \cdot |T|)}, where cgc_g is a constant depending solely on the genus of GG and ff is a computable function. We complement this result by showing that there is no f(R)V(G)o(T2/logT)f(R)\cdot |V(G)|^{o(|T|^2/ \log |T|)} algorithm for any function ff for the problem on general graphs, unless ETH fails
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